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Angular speed is a measure of how quickly an object rotates or revolves relative to another point, often the center of a circular path. It is designated by the Greek letter \(\omega\). Angular speed is determined by dividing the total angle of rotation by the time it takes to cover that angle. The formula is:

• \(\omega = \frac{\theta}{t}\)

where:

• \(\omega\) is the angular speed in radians per second
• \(\theta\) is the angular displacement in radians
• \(t\) is the time in seconds

In our exercise, to find the angular speed, we calculated the time it takes for the baseball to travel the horizontal distance and used the given angular displacement. This allowed us to determine how fast the baseball was spinning around its axis, giving us an angular speed of approximately \(126.29\ \text{rad/s}\). This concept is crucial in understanding rotational motion.

Tangential speed is the speed of something moving along a circular path. It refers to how fast a point on a rotating object is moving in a direction tangent to the circle. The tangential speed can be found using the formula:

• \(v_t = \omega \times r\)

where:

• \(v_t\) is the tangential speed in meters per second
• \(\omega\) is the angular speed
• \(r\) is the radius of the circular path

In this baseball exercise, the tangential speed was calculated for a point on the "equator" of the baseball. This was achieved by multiplying the angular speed by the radius of the baseball. It resulted in a tangential speed of approximately \(4.63\ \text{m/s}\). This indicates how fast a point on the surface of the baseball is traveling as it spins.

Conversions are often necessary in physics to ensure that the units align for calculations. In the context of circular motion, the radius must be in meters if we're working in the metric system, as this helps maintain consistency in units such as meters per second for speed. To convert the radius from centimeters to meters, you should:

• Use the conversion factor: \(1\ \text{m} = 100\ \text{cm}\)

For example, given a radius of \(3.67\ \text{cm}\):

• \(3.67\ \text{cm} \times \frac{1 \text{m}}{100 \text{cm}} = 0.0367\ \text{m}\)

This was the first step in solving our exercise. Ensuring the radius is in meters allows for correct and straightforward application of formulas involving radius, such as when calculating tangential and angular speed.

Angular displacement is the measure of the angle through which an object rotates around a particular point or axis. It is usually measured in radians, which is the standard unit of angular displacement in the metric system.

• Angular displacement represents the cumulative angle turned, unlike linear displacement which reflects distance covered.

In the exercise, the baseball rotates by an angle of \(49.0\ \text{rad}\) as it travels downfield. This angular displacement is crucial when calculating angular speed, as it describes the total rotation relative to a given point within a specific timeframe. Understanding angular displacement helps us comprehend how far an object has turned from its initial position.